We propose a simple derivation of the one-dimensional hard-rod equation of state, with and without a Kac tail (a long-range and weak potential). The case of hard spheres in higher dimension is also addressed, and we recover the virial form of the equation of state in a direct way.
REFERENCES
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For a thorough historical overview, see
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See for example,
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To be more specific, we need to introduce reduced coordinates and reduced lengths . The Hamiltonian expressed in terms of becomes -independent so that the canonical partition function is , For interactions other than hard core, this scaling relation for the partition function still holds, which immediately leads to Eq. (3).
13.
M.
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The fundamental difference between and higher space dimensions is that in one dimension because the particle-particle pair distribution function at contact coincides with , the particle-wall distribution function at contact. For , which differs from .
15.
Another method, due to H. S. Green, may be found in
T. L.
Hill
, Statistical Mechanics
(Dover
, New York, 1987
), Chap. 6. This method relies on a general rescaling of particle sizes with system size, and is therefore reminiscent of the argument backing up Eq. (3), with the difference that in calculating the volume derivative of the partition function, the particle sizes are kept fixed.16.
J.
Zhang
, R.
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, E.
Trizac
, J. A.
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, and D.
Frenkel
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Formally, the limit , with fixed is required, and should be taken after the thermodynamic limit (Refs. 6 and 7).
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).© 2008 American Association of Physics Teachers.
2008
American Association of Physics Teachers
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